1 Cultural Connection The Revolt of the Middle Class Student led discussion. The Eighteenth Century in Europe and America.

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Presentation on theme: "1 Cultural Connection The Revolt of the Middle Class Student led discussion. The Eighteenth Century in Europe and America."— Presentation transcript:

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1 Cultural Connection The Revolt of the Middle Class Student led discussion. The Eighteenth Century in Europe and America.

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2 12 – The 18 th Century and Exploitation of the Calculus The student will learn about More great mathematicians and the advances of applied mathematics.

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3 §12-1 Introduction Student Discussion.

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4 §12-1 Introduction Fifty percent of all known mathematics was created during the past fifty years, and fifty percent of all mathematicians who have ever lived are alive today. Before periodicals containing some math. 18 th Century210 periodicals containing some math. 19 th Century950 periodicals containing only math. 20 th Century2600 periodicals containing only math. Growth of insurance, economics, technology problems, industrialization, world wars, computers, space program – applied mathematics.

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5 §12-2 The Bernoulli’s Student Discussion.

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6 §12-2 The Petersburg Paradox Nicolaus Bernoulli. If A receives a penny when a head appears on the first toss of a coin, two pennies if a head does not appear until the second toss, four pennies if a head does not appear until the third toss, and so on, what is A’s expectation? EventPValueExpect H½1 ¢½ ¢ TH¼2 ¢½ ¢ TTH1/84 ¢½ ¢ TTTH1/168 ¢½ ¢ Σ = 

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12 §12- 4 Taylor & Maclaurin Series The Maclaurin and Taylor series are polynomials used to approximate functions. Taylor Series - Maclaurin Series – c = 0 in Taylor OR Note that this diverges rather quickly because of the denominator of k!.

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20 §  is irrational If x  0 is rational, then tan x is irrational. The contra positive of the above is then logically true, or if tan x is rational then x is irrational. But tan  /4 = 1 so  /4 is irrational and hence  is irrational. Johann Lambert